It really isn't that hard. With modern electronics it isn't hard to make very accurate time measurements. The simple answer is you take a very fast clock and count the number of clock pulses that take place between events. If the clock is say 20GHz it will through 20 pulses in one nanosecond. So you set up the measurement to start when you send the particle and stop when the particle arrive then you look at how many clock pulses took place. You know the speed of light and how far you are sending so if it gets there 5 pulses faster than light would have taken then it happened in one quarter of a nanosecond or 250 picoseconds faster than the speed of light.

What they are doing is much more complicated than that but that is a 3000 foot level view of how you could make the measurement.